M. Masili, ICTP

In this lesson continues with the study of **canonical ensemble**. The canonical ensemble is a specific **statistical ensemble** in which the probability of finding a particular microstate *X*_{n} with energy *E _{n}* follows the

**Boltzmann distribution**

*P(X _{n}) = Z^{-1} *exp{

*-?E*}

_{n}where *Z ^{-1}* is a normalization constant. The quantity

*Z*is called the

**partition function**and is given by

*Z = ? *exp{*-?E _{n}*}

where the sum runs over all microstates of the system.

The canonical ensemble gives the statistical description of a system in thermal contact with a heat bath at temperature *T*. The quantity *?* is determined by the temperature of the bath: *? = (kT)*^{-1}, *k* is the **Boltzmann constant**.

A key fact, and the source of the power of of statistical mechanics, is that the partition function encodes all the thermodynamics of the system. All thermodynamical quantities such as the energy, entropy, and the rest of the potentials can be directly obtained if *Z* is known. Thus, if the energy levels of a system are known (microscopic), a simple algorithm for computing the thermodynamical quantities (macroscopic) ensues. The fundamental problem of statistical mechanics is therefore reduced to calculating the partition function of the system.

Using the canonical partition function, an expression for entropy was derived

*S = ? P*_{n} log{*P _{n}*}

This expression is actually more general and gives a measure of the uncertainty about the microscopic state of the system once the macroscopic state has been fixed (for instance, specifying *N*, *V* and *T*). It reduces to the **Boltzmann formula for the entropy** when the microstates are equidistributed.

Finally, all this machinery was put to use to calculate the partition function and thermodynamical quantities for the ideal gas. The thermodynamics was found to be the same as the one found in the microcanonical ensemble.

As a complementary tool you can also see some lessons onĀ Statistical Mechanics given in theĀ

University.Lecture 6 | Modern Physics: Statistical MechanicsMay 4, 2009 - Leonard Susskind explains the second law of thermodynamics, illustrates chaos, and discusses how the volume of phase space grows.